Applied Mathematics
Vol.10 No.10(2019), Article ID:95429,12 pages
10.4236/am.2019.1010057
On the First and Second Locating Zagreb Indices of Graphs
Suha A. Wazzan1*, Anwar Saleh2
1Department of Mathematics, Faculty of Science, KAU King Abdulaziz University, Girls Campus, Jeddah, KSA
2Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, KSA
Copyright © 2019 by author(s) and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: September 7, 2019; Accepted: September 24, 2019; Published: September 27, 2019
ABSTRACT
By the distance or degree of vertices of the molecular graph, we can define graph invariant called topological indices. Which are used in chemical graph to describe the structures and predicting some physicochemical properties of chemical compound? In this paper, by introducing two new topological indices under the name first and second Zagreb locating indices of a graph G, we establish the exact values of those indices for some standard families of graphs included the firefly graph.
Keywords:
Zagreb Indices, First and Second Locating Indices, Join Graph, Firefly Graph
1. Introduction
Topological indices play a significant role mainly in chemistry, pharmacology, etc. (see [1] - [7] ). Many of the topological indices of current interest in mathematical chemistry are defined in terms of vertex degrees of the molecular graph. Two of the most famous topological indices of graphs are the first and second Zagreb indices which have been introduced by Gutman and Trinajstic in [8] , and defined as and , respectively. The Zagreb indices have been studied extensively due to their numerous applications in the place of existing chemical methods which need more time and increase the costs. Many new reformulated and extended versions of the Zagreb indices have been introduced for several similar reasons (cf. [9] - [17] ).
One of the present authors Saleh [18] has recently introduced a new matrix representation for a graph G by defining the locating matrix over G. We will redefine this representation as in the following.
Definition 1 ( [18] ) Let be a connected graph with vertex set . A locating function of G denoted by is a function such that , where is the distance between the vertices and in G. The vector is called the locating vector corresponding to the vertex , where is actually the dot product of the vectors and in the integers space such that is adjacent to .
The above locating function and huge applications of Zagreb indices motivated us to introduce two new topological indices, namely first and second locating indices, based on the locating vectors.
Definition 2. Let be a connected graph with a vertex set and an edge set . Then we define the first and second locating indices as
respectively.
All graphs in this paper will be assumed simple, undirected and connected unless stated otherwise. For graph theoretical terminologies, we refer [19] to the readers.
2. Some Exact Values in Terms of Locating Indices
In this section, by considering Definition 2, we determine the first and second locating indices for the standard graphs ,,,,, and also for the join graph such that and are both connected graphs with diameter 2 and G will be assumed as , -free graphs.
Theorem 3. Let be the complete graph with a vertex set , where . Then and .
Proof. Let be a locating vector corresponding to the vertex . Then such that and . Thus . But we have total n vertices in , and so , as required. On the other hand, for any two locating vectors and , where , we
definitely have . Hence .+
In the next two Theorems, we investigate the cycle depends on the status of n.
Theorem 4. For an even integer , let . Then and .
Proof. By labeling the vertices of the cycle as in the anticlockwise direction, we obtain
and hence . It is not difficult to see that each has the same components within different location, and so each has the same sum as the form of . Therefore . In addition, by the symmetry,
which gives .+
Theorem 5. For an odd integer , let . Then and .
Proof. With a similar procedure as in the proof of Theorem 4, we get
which implies
and so . Also, by the symmetry,
which gives the exact value of as depicted in the statement of theorem.+
Now we will take into account the complete bipartite graphs to determine the locating indices.
Theorem 6. Let , where . Then and .
Proof. For all and , by labeling the adjacent vertices and of , the locating vectors of are given by:
In here, for any , we have and for any , we get . Therefore
On the other hand, for any two consecutive locating vertices in , since , we obtain .+
Since the following consequences of Theorem 6 are very special cases and clear, we will omit their proofs.
Corollary 7. Let , where . Then and .
Corollary 8. Let . Then and .
The case for wheel graphs will be investigated in the following result.
Theorem 9. Let us consider G as the wheel graph ( ) with vertices. Then we have and .
Proof. With a similar approximation as in the previous results, by labeling the vertices of in the anticlockwise direction as such that is the center of the wheel, we obtain
Now for any locating vector corresponding to a vertex , we have and . Hence .
For , by labeling the vertices as above, we have
Bearing in mind the permutation of components in each vector , where , it is easy to see that any two adjacent vertices and satisfy and for . Hence .+
The result for determining of locating indices on path graphs can be given as in the following.
Theorem 10. Let . Then
and
Proof. Assume that G is the graph ( ). By labeling the vertices from left to right as according to the locating function, the corresponding vector for each vertex ( ) will be the form of
By applying the symmetry on components between the vector pairs and and so on, we can see that
For , we see that
However, by the symmetry between the components of the vectors as mentioned above, we get
which can be rewritten as in the form
This complete the proof.+
It is known that from the elementary textbooks the join of graphs and with disjoint vertex sets and and edge sets and is the graph union together with all the edges joining and . In the following theorem we find first and second locating indices for the join graph G.
Theorem 11. Let such that and are both connected graphs with diameter 2 and G is a or -free graph. Assume that has vertices and edges while has vertices and edges. Then
and
Proof. Assume that G satisfies the conditions in the statement of theorem. Let us label the vertices of the graph G as
where and . Also let be the locating vector corresponding to the vertex v such that :
Then .
Similarly, for any vertex , the locating vector corresponding to w:
So . Therefore, by the above equalities on and , we obtain
Now, let us make partition to the set of vertices of G as
Hence can be written as . To get for any two adjacent vertices , let us consider
We then have
which implies . With a similar calculation, we get .
Next, we need to calculate . To do that let us take and , and then labeling as
Hence we get
and so .
After all above calculations, we finally obtain
Hence the result.+
3. Locating Indices of Firefly Graphs
We recall that a firefly graph ( and ) is a graph of order n that consists of s triangles, t pendant paths of length 2 and pendent edges that are sharing a common vertex (cf. [20] ). Let be the set of all firefly graphs . Note that contains the stars , stretched stars , friendship graphs and butterfly graphs .
In the next theorem we present the first and second locating indices for the firefly graph. In our calculations, for simplicity, we denote by a single letter l.
Theorem 12. Let ( and ) be a firefly graph of order n. Then
and
Proof. Let ( and ) is a firefly graph of order n. Let us label the vertices with clockwise direction as
where is the center of the firefly graph and
Now we calculate the corresponding vectors for each vertex , where , as in the following:
Suppose that such that
Therefore we can write
For the calculation of , we have the cases and
where . Hence
On the other hand, for the calculation of , we have
where for . Thus
Thirdly to calculate , we have
where , and so
Finally, for the case of , we get
where . This gives
By collecting all above calculations, we obtain
as required.
Before starting to calculate the index , we should remind that for any two adjacent vertices u and v will be denoted by . Now, let us again consider the same subsets A, B, C and D of . Therefore we firstly have
Secondly,
Thirdly,
Again, by collecting all above calculations, we obtain
These all above processes complete the proof.+
Corollary 13. 1) For any friendship graph of order n,
2) For any butterfly graph of order n,
4. Conclusion
In this paper, two new topological indices based on Zagreb indices are proposed. The exact values of these new topological indices are calculated for some standard graphs and for the firefly graphs. These new indices can be used to investigate the chemical properties for some chemical compound such as drugs, bridge molecular graph etc. For the future work, instead of defining these new topological indices based on the degrees of the vertices, we can redefine them based on the degrees of the edges by defining them on the line graph of any graph. Similar calculations can be computed to indicate different properties of the graph.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Cite this paper
Wazzan, S.A. and Saleh, A. (2019) On the First and Second Locating Zagreb Indices of Graphs. Applied Mathematics, 10, 805-816. https://doi.org/10.4236/am.2019.1010057
References
- 1. Diudea, M.V., Florescu, M.S. and Khadikar, P.V. (2006) Molecular Topolgy and Its Applications. Eficon, Bucarest.
- 2. Diudea, M.V. (2010) Nanomolecules and Nanostructures-Poynomial and Indices. Univ. Kragujevac, Kragujevac.
- 3. Gutman, I. and Furula, B. (2012) Distance in Molecular Graphs. Univ. Kragujevac, Kragujevac.
- 4. Gutman, I. and Furula, B. (2010) Novel Molecular Structure Descriptors-Theory and Applications II. Univ. Kragujevac, Kragujevac.
- 5. Karelson, M. (2000) Molecular Descriptors in QSAR-QSPR. Wiley, New York.
- 6. Todeschini, R. and Consonni, V. (2000) Handbook of Molecular Descriptors. Wiley-VCH, Weinheim. https://doi.org/10.1002/9783527613106
- 7. Todeschini, R. and Consonni, V. (2009) Molecular Descriptors for Chemoinformatics. Wiley-VCH, Weinheim, Vols. I and II. https://doi.org/10.1002/9783527628766
- 8. Gutman, I. and Trinajstic, N. (1972) Graph Theory and Molecular Orbitals, Total π-Electron Energy of Alternant Hydrocarbons. Chemical Physics Letters, 17, 535-538. https://doi.org/10.1016/0009-2614(72)85099-1
- 9. Das, K.C., Yurttas, A., Togan, M., Cevik, A.S. and Cangul, I.N. (2013) The Multiplicative Zagreb Indices of Graph Operation. Journal of Inequalities and Applications, 2013, Article No. 90. https://doi.org/10.1186/1029-242X-2013-90
- 10. Alwardi, A., Alqesmah, A., Rangarajan, R. and Cangul, I.N. (2018) Entire Zagreb Indices of Graphs. Discrete Mathematics, Algorithms and Applications, 10, Article ID: 1850037. https://doi.org/10.1142/S1793830918500374
- 11. Braun, J., Kerber, A., Meringer, M. and Rucker, C. (2005) Similarity of Molecular Descriptors: The Equivalence of Zagreb Indices and Walk Counts. MATCH Communications in Mathematical and in Computer Chemistry, 54, 163-176.
- 12. Gutman, I. and Das, K.C. (2004) The First Zagreb Index 30 Years after. MATCH Communications in Mathematical and in Computer Chemistry, 50, 83-92.
- 13. Khalifeh, M.H., Youse-Azari, H. and Ashra, A.R. (2009) The First and Second Zagreb Indices of Some Graph Operations. Discrete Applied Mathematics, 157, 804-811. https://doi.org/10.1016/j.dam.2008.06.015
- 14. Nikolić, S., Kovacević, G., Milicević, A. and Trinajstić, N. (2003) The Zagreb Indices 30 Years after. Croatica Chemica Acta, 76, 113-124.
- 15. Zhou, B. and Gutman, I. (2005) Further Properties of Zagreb Indices. MATCH Communications in Mathematical and in Computer Chemistry, 54, 233-239.
- 16. Zhou, B. and Gutman, I. (2004) Relations between Wiener, Hyper-Wiener and Zagreb Indices. Chemical Physics Letters, 394, 93-95. https://doi.org/10.1016/j.cplett.2004.06.117
- 17. Zhou, B. (2004) Zagreb Indices. MATCH Communications in Mathematical and in Computer Chemistry, 52, 113-118.
- 18. Ramaswamy, H.N., Alwardi, A. and Kumar, N.R. (2017) On the Locating Matrix of a Graph and Its Spectral Analysis. Computer Science Journal of Moldova, 25, 260-277.
- 19. Chartand, G. and Lesniak, L. (2005) Graphs and Digraphs. 4th Edition, CRC Press, Boca Raton.
- 20. Li, J.X., Guo, J.M. and Shiu, W.C. (2013) On the Second Largest Laplacian Eignvalues of Graphs. Linear Algebra and Its Applications, 438, 2438-2446. https://doi.org/10.1016/j.laa.2012.10.047